A062051 Number of partitions of n into powers of 3.
1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 23, 23, 23, 28, 28, 28, 33, 33, 33, 40, 40, 40, 47, 47, 47, 54, 54, 54, 63, 63, 63, 72, 72, 72, 81, 81, 81, 93, 93, 93, 105, 105, 105, 117, 117, 117, 132, 132, 132, 147, 147, 147, 162
Offset: 0
Keywords
Examples
a(4) = 2 and the partitions are 3+1, 1+1+1+1; a(9) = 5 and the partitions are 9; 3+3+3; 3+3+1+1+1; 3+1+1+1+1+1+1; 1+1+1+1+1+1+1+1+1.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
- Vassil Dimitrov, Laurent Imbert, and Andrew Zakaluzny, Multiplication by a Constant is Sublinear, 18th IEEE Symposium on Computer Arithmetic (2007). See Theorem 1.
- Md Towhidul Islam and Md Shahidul Islam, Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance, arXiv:1502.07730 [math.CO], 2015.
- M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
- M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]
Crossrefs
Programs
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Mathematica
nn=70;a=Product[1/(1-x^(3^i)),{i,0,4}];CoefficientList[Series[a,{x,0,nn}],x] (* Geoffrey Critzer, Oct 30 2012 *) -
PARI
{ n=15; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*3)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry -
Python
from functools import lru_cache @lru_cache(maxsize=None) def A062051(n): return A062051(n-1)+(0 if n%3 else A062051(n//3)) if n>2 else 1 # Chai Wah Wu, Sep 21 2022
Formula
a(n) = A005704([n/3]).
G.f.: Product_{k>=0} 1/(1-x^(3^k)). - R. J. Mathar, Jul 31 2010
If m = ceiling(log_3(2k)), define n = (3^m + 1)/2 - k for k in the range (3^(m-1)+1)/2 + (3^(m-2)) <= k <= (3^m-1)/2. Then, a(n) = Sum_{s=ceiling((k-1)/3)..(3^(m-1)-1)/2} a(s). This gives the first 2(3^(m-1))/3 terms. - Md. Towhidul Islam, Mar 01 2015
G.f.: 1 + Sum_{i>=0} x^(3^i) / Product_{j=0..i} (1 - x^(3^j)). - Ilya Gutkovskiy, May 07 2017
Extensions
More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001
Comments